The Algebraic Hodge Theorem and the Fundamental Theorem of Elliptic Operators

Definition. Let and be inner product spaces. Let be a linear map. Then a linear map is called the formal adjoint of if for any and any .

Lemma. (1) If a formal adjoint exists, it is unique. (2) If , then exists.

Example. The map defined by summing the coodinates has no formal adjoint, where is the colimit of .

If and are Hilbert spaces, then we have

Theorem. Any continuous linear map of Hilbert spaces has a formal adjoint.

Example. Let be a differential operator. Suppose and have smooth inner product structures, then we have the “-inner products” on and , given by . Then has a formal adjoint . If we write locally as , then .

An elliptic operator is self-adjoint if .

We can now state the Fundamental Theorem of Elliptic Operators. Later in this entry we will give some corollaries and much later in the course we will outline a proof using the method of elliptic regularity.

Fundamental Theorem of Elliptic Operators. For a self-adjoint elliptic operator , there is an orthogonal decomposition with finite-dimensional.

It is important here that the manifold is closed.

The algebraic Hodge theorem

Suppose now we have (co)chain complex over or :

Give each an inner product. Assume each has a formal adjoint . Define the Laplacian . Then we have

Lemma. iff and .

Proof. Suppose . Then we have

and hence .

Theorem. Let be a (co)chain complex over a field . Then there exist decompositions such that the (co)chain complex can be written as

.

When or and is finite dimensional for each , setting and , the theorem above becomes a corollary of the following Algebraic Hodge Theorem:

Algebraic Hodge Theorem. Let be a (co)chain complex over or . Suppose that has inner product for each and that formal adjoint exists for each . Let , then

(1) TFAE: (a) , (b) and , (c) .

(2) .

(3) If is finite dimensional, then .

(4) If for any , then there are orthogonal decompositions

Proof. (1) (a) (b) (c) (a).

(2) Let , then .

(3) Show the inclusion in (2) is an equality by counting dimensions.

(4) It suffices to show the following orthogonal decomposition:

However easily we have

By checking the decomposition diagram above, we can obtain:

Corollary. is an isomorphism, where

Corollary. iff is an isomorphism for any .

Wrapping up

Corollary. Algebraic Wrapping up. For as above: is an isomorphism. Hence is an isomorphism for all p.

This corollary “wraps up” a (co)chain complex into a single map.

Next, we consider wrapping up an ellliptic complex.

Definition. An elliptic complex of differential operators is a cochain complex of differential operators

so that for all the associated symbol complex is exact.

If we define the symbol of differential operator of order by , then we have .

Proposition. Let be an elliptic complex of differential operators. Give and metrics for each . Then is an elliptic operator.

Proof. For any , since the complex is elliptic, we have the exact sequence

Thus, is an isomorphism.

Finally note that .

Consequences of the fundamental theorem.

We deduce the following corollaries of the Fundamental Theorem, the Algebraic Hodge Theorem, and Wrapping Up.

Corollary. Let be an elliptic complex of differential operators.

(1) For any , is finite dimensional.

(2) For any , .

(3) .

Corollary. If is an elliptic differential operator, then we have isomorphisms . Hence the kernel and the cokernel of an elliptic differential operator are finite dimensional.